Dynamics of Partial Differential Equations

Volume 21 (2024)

Number 1

On the well-posedness in Besov–Herz spaces for the inhomogeneous incompressible Euler equations

Pages: 1 – 29

DOI: https://dx.doi.org/10.4310/DPDE.2024.v21.n1.a1

Authors

Lucas C. F. Ferreira (Department of Mathematics, State University of Campinas, SP, Brazil)

Daniel F. Machado (Department of Mathematics, State University of Campinas, SP, Brazil)

Abstract

In this paper we study the inhomogeneous incompressible Euler equations in the whole space $\mathbb{R}^n$ with $n \geq 3$. We obtain well-posedness and blow-up results in a new framework for inhomogeneous fluids, more precisely Besov–Herz spaces that are Besov spaces based on Herz ones, covering particularly critical cases of the regularity. Comparing with previous works on Besov spaces, our results provide a larger initial data class for a well-defined flow. For that, we need to obtain suitable linear estimates for some conservation-law models in our setting such as transport equations and the linearized inhomogeneous Euler system.

Keywords

inhomogeneous Euler equations, well-posedness, blow-up criterion, transport equations, commutator estimates, Besov–Herz spaces

2010 Mathematics Subject Classification

35B30, 35Q31, 35Q35, 42B35, 42B37, 76B03

Received 26 April 2023

Published 7 November 2023